how to know which numbers to multiply from in Elimination Method

Question

When solving linear equations sometimes the equations are multiplied with different numbers to cancel out the variable.. so my question is how to know which number to multiply the equation with??

Answer 1

suppose 2x + 3y = 11...Eq. A

             5x + 7y = 23...Eq. B

 

This example will be sufficient to illustrate the general idea. Observe that the coefficients of x and y are all different prime numbers; our task will be to choose some number(s) so that new versions of A and B, when added, will cause one of the terms (x or y) to disappear.

 

Let's try to make y disappear.  We will multiply A by 7 and B by -3, to obtain

A:  14x + 21y = 77

B:  -15x - 21y = -69

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sum: -x          = 8   so x = -8

 

Finding y is easy; just plug -8 into one of the original equations.

 

You can always eliminate one of x or y by multiplying by appropriate choices (essentially swapping coefficients as multipliers).